A note on the multiplicative group of a division ring

For any group $G$G its FC-centre may be defined as the subset of all elements having only finitely many conjugates. This is a subgroup and it allows one to define the upper central FC-series $\{F_n\}${Fn} by $F_0=1,\ F_1={\rm FC}(G)$F0=1, F1=FC(G) and $F_{n+1}/F_n={\rm FC}(G/F_n)$Fn+1/Fn=FC(G/Fn). If $F_n=G$Fn=G for some $n$n, then $G$G is said to be FC-nilpotent. The author shows that if $D$D is a division ring with centre $Z$Z and $x\in D$x∈D has infinitely many conjugates, then so does its image $xZ^\times$xZ× in $D^\times/Z^\times$D×/Z×, where $D^\times$D× is the multiplicative group of $D$D. He deduces that any division ring $D$D whose group $D^\times$D× is FC-nilpotent is commutative, thus generalizing a theorem which asserts commutativity when $D^\times$D× is nilpotent [L.-K. Hua, Acad. Sinica Science Record 3 (1950), 1--6; MR0039707 (12,584e)]. It follows that $D$D is commutative whenever $D^\times$D× has the permutation property ${\rm P}_n$Pn defined by A. Restivo and C. Reutenauer [J. Algebra 89 (1984), no. 1, 102--104; MR0748230 (85k:20188)].